Historical context
Dr Christopher Hollings, Departmental Lecturer in Mathematics and its History, has written a very interesting two-page document picking out themes and people relevant to the course. I strongly recommend reading it for an introduction to the historical context of this course, and it gives lots of suggestions of follow-up reading.
The MacTutor website is always worth consulting on the history of mathematics. It has biographies of many, many mathematicians, and also articles on historical topics. There's even a list of historical topics relating to analysis. For example, this one on 'The real numbers: Stevin to Hilbert caught my eye in the context of Analysis I (it explores mathematicians' understanding over a period of time of what real numbers are, and what properties they have).
The book A radical approach to real analysis by David M. Bressoud introduces the ideas of our Analysis I course, and more, with a historical perspective showing how and why the definitions and results of analysis emerged and evolved. It's available electronically via the university, see the Reading List.